St\"ackel and Eisenhart lifts, Haantjes geometry and Gravitation

We study lifts of integrable systems by means of generalized St\"ackel geometry. To this aim, we present the notion of St\"ackel lift as a unified setting for the construction of new classes of integrable Hamiltonian systems of physical interest. The St\"ackel lift extends the geometric framework underlying both the Riemannian and the Lorentzian-type classical Eisenhart lifts. Moreover, we prove that Hamiltonian systems constructed through momentum-dependent St\"ackel matrices are naturally endowed with a non-trivial symplectic-Haantjes structure. We further illustrate applications to magnetic systems separable in cylindrical coordinates; we describe them within the St\"ackel framework by means of modified St\"ackel basis. We also show that explicitly momentum-dependent lifting matrices produce systems interpretable as gravitational waves, or momentum-dependent metrics of Hamilton and Finsler geometries, with potential applications in modified gravity theories.